Potential of two charges at the midpoint

Problem

Two point charges $q_1 = +4\;\mu\text{C}$ and $q_2 = -2\;\mu\text{C}$ are fixed at points $A = (0,0)$ and $B = (0{,}20;\,0)\;\text{m}$. Calculate the electric potential at point $P = (0{,}10;\,0)\;\text{m}$ (the midpoint of the segment), taking $V = 0$ at infinity.

Solution

Superposition principle. Potential is a scalar quantity: at point $P$ it is the algebraic sum (with sign) of the potentials due to the individual charges. $V_P = k\,\frac{q_1}{r_1} + k\,\frac{q_2}{r_2}$

Distances. Point $P$ is the midpoint, so it is equidistant from the two charges: $r_1 = r_2 = 0{,}10\;\text{m}$

Numerical substitution. $V_P = (9\cdot 10^9)\left(\frac{4\cdot 10^{-6}}{0{,}10} + \frac{-2\cdot 10^{-6}}{0{,}10}\right) = (9\cdot 10^9)\,\frac{2\cdot 10^{-6}}{0{,}10}$

Result. $\ev{V_P \approx 1{,}8\cdot 10^{5}\;\text{V}}$

The resulting potential is positive because the positive charge prevails in magnitude; note that, unlike the field (a vector), here the contributions simply add up with their sign.

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Topics: Campo elettrico e potenziale Concepts: Potenziale elettrico Skills: Principio di sovrapposizione Objects: Sfera carica